🤖 AI Summary
This paper investigates the construction of e-variables and e-processes under composite exponential family null hypotheses. It systematically compares four approaches: reverse information projection (RIPr), conditional likelihood ratio (COND), universal inference (UI), and sequential RIPr. The work establishes, for the first time, the exact form of the RIPr prior in the Gaussian case and derives necessary and sufficient conditions for equivalence between RIPr and COND e-variables. Theoretically, it reveals a $(d/2)log n$ efficiency loss for UI and rigorously proves that COND is optimal in e-power. Precise expressions for e-power are derived for Gaussian models, and $o(1)$-accurate approximations are provided for general exponential families. The core contribution is a unifying framework that clarifies relationships among these methods and establishes COND as both theoretically optimal and practically implementable for e-variable construction.
📝 Abstract
We analyze common types of e-variables and e-processes for composite exponential family nulls: the optimal e-variable based on the reverse information projection (RIPr), the conditional (COND) e-variable, and the universal inference (UI) and sequen-tialized RIPr e-processes. We characterize the RIPr prior for simple and Bayes-mixture based alternatives, either precisely (for Gaussian nulls and alternatives) or in an approximate sense (general exponential families). We provide conditions under which the RIPr e-variable is (again exactly vs. approximately) equal to the COND e-variable. Based on these and other interrelations which we establish, we determine the e-power of the four e-statistics as a function of sample size, exactly for Gaussian and up to $o(1)$ in general. For $d$-dimensional null and alternative, the e-power of UI tends to be smaller by a term of $(d/2) log n + O(1)$ than that of the COND e-variable, which is the clear winner.